## Topological Methods in Nonlinear Analysis

### Realization of a graph as the Reeb graph of a Morse function on a manifold

Łukasz Patryk Michalak

#### Abstract

We investigate the problem of the realization of a given graph as the Reeb graph $\mathcal{R}(f)$ of a smooth function $f\colon M\rightarrow \mathbb{R}$ with finitely many critical points, where $M$ is a closed manifold. We show that for any $n\geq2$ and any graph $\Gamma$ admitting the so-called good orientation there exist an $n$-manifold $M$ and a Morse function $f\colon M\rightarrow \mathbb{R}$ such that its Reeb graph $\mathcal{R}(f)$ is isomorphic to $\Gamma$, extending previous results of Sharko and Masumoto-Saeki. We prove that Reeb graphs of simple Morse functions maximize the number of cycles. Furthermore, we provide a complete characterization of graphs which can arise as Reeb graphs of surfaces.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 2 (2018), 749-762.

Dates
First available in Project Euclid: 9 August 2018

https://projecteuclid.org/euclid.tmna/1533780029

Digital Object Identifier
doi:10.12775/TMNA.2018.029

Mathematical Reviews number (MathSciNet)
MR3915662

Zentralblatt MATH identifier
07051691

#### Citation

Michalak, Łukasz Patryk. Realization of a graph as the Reeb graph of a Morse function on a manifold. Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 749--762. doi:10.12775/TMNA.2018.029. https://projecteuclid.org/euclid.tmna/1533780029

#### References

• S. Biasotti, D. Giorgi, M. Spagnuolo and B. Falcidieno, Reeb graphs for shape analysis and applications, Theoret. Comput. Sci. 392 (2008), 5–22.
• K. Cole-McLaughlin, H. Edelsbrunner, J. Harer, V. Natarajan and V. Pascucci, Loops in Reeb graphs of $2$-manifolds, Discrete Comput. Geom. 32 (2004), 231–244.
• I. Gelbukh, Loops in Reeb graphs of $n$-manifolds, Discrete Comput. Geom. 59 (2018), no. 4, 843–863.
• I. Gelbukh, The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations, Math. Slovaca 67 (2017), 645–656.
• M. Kaluba, W. Marzantowicz and N. Silva, On representation of the Reeb graph as a sub-complex of manifold, Topol. Methods Nonlinear Anal. 45 (2015), no. 1, 287–307.
• M.A. Kervaire and J.W. Milnor, Groups of Homotopy Spheres I, Ann. of Math. (2), vol. 77, no. 3. (May, 1963), 504–537.
• J. Martinez-Alfaro, I.S. Meza-Sarmiento and R. Oliveira, Topological classification of simple Morse Bott functions on surfaces, Contemp. Math. 675 (2016), 165–179.
• Y. Masumoto and O. Saeki, A smooth function on a manifold with given Reeb graph, Kyushu J. Math. 65 (2011), no. 1, 75–84.
• J.W. Milnor, Differential Topology, Lectures on Modern Mathematics, Vol. II, Wiley, New York, 1964, 165–183.
• J.W. Milnor, Lectures on the h-Cobordism Theorem, Princeton University Press, Princeton, 1965.
• G. Reeb, Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique, C.R. Acad. Sci. Paris 222 (1946), 847–849.
• V.V. Sharko, About Kronrod–Reeb graph of a function on a manifold, Methods Funct. Anal. Topology 12 (2006), 389–396.
• F. Takens, The minimal number of critical points of a function on a compact manifold and the Lusternik–Schnirelman category, Invent. Math. 6 (1968), 197–244.